Volume Comparison Theorem Research Articles

Volume comparison theorems in Finsler geometry

Volume comparison theorems in Finsler geometry

Extension of Bishop–Gromov Volume Comparison Theorem by Elliptic Operators

Extension of Bishop–Gromov Volume Comparison Theorem by Elliptic Operators

Scalar curvature volume comparison theorems for almost rigid spheres

Bray’s football theorem gives a sharp volume upper bound for a three dimensional manifold with scalar curvature no less than n ( n − 1 ) n(n-1) and Ricci curvature at least ε 0 g ¯ \varepsilon _0 \bar {g} . This paper extends Bray’s football theorem in higher dimensions, assuming the manifold is axisymmetric or the Ricci curvature has a uniform upper bound. Effectively, we show that if the Ricci curvature of an n n -manifold is close to that of a round n-sphere, a lower bound on scalar curvature gives an upper bound on the total volume.

Volume comparison theorems in Finsler spacetimes

In a [Formula: see text]-dimensional Lorentz–Finsler manifold with [Formula: see text]-Bakry–Émery Ricci curvature bounded from below, where [Formula: see text], using the Riccati equation techniques, we establish the Bishop–Gromov volume comparison theorem for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs). We also establish the Günther volume comparison theorem for SCLVs when the flag curvature is bounded above.

Uniform Poincaré inequalities on measured metric spaces

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincar\'e inequalities on $(X,d,\mu)$ if it satisfies a local Poincar\'e inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a $(\sigma,\beta,\sigma)$-Poincar\'e inequality. If $(X,d,\mu)$ is a $\delta$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincar\'e inequality with exponential growth of the Poincar\'e constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space then it supports a uniform Poincar\'e inequality and the Poincar\'e constant depends on the growth of the fundamental group.

New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds

Let L = Δ − ⟨ ∇ ϕ , ∇ · ⟩ $L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure μ ( d x ) = e − ϕ ( x ) m ( d x ) $\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$ on a complete non-compact smooth Riemannian manifold ( M , g ) $(M,g)$ with its volume element m = vol g $\mathfrak {m} =\text{\rm vol}_g$ , and ϕ ∈ C 2 ( M ) $\phi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with CD ( K , m ) ${\rm CD}(K, m)$ -condition for m ⩽ 1 $m\leqslant 1$ and a continuous function K $K$ . As consequences, we give the optimal conditions on m $m$ -Bakry–Émery Ricci tensor for m ⩽ 1 $m\leqslant 1$ such that the (weighted) Myers' theorem, Bishop–Gromov volume comparison theorem, stochastic completeness and Feller property of L $L$ -diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well studied for m $m$ -Bakry–Émery Ricci curvature for m ⩾ n $m\geqslant n$ (Li, J. Math. Pures Appl. (9) 84 (2005), 1295–1361; Lott, Comment. Math. Helv. 78 (2003), 865–883; Qian, Q. J. Math. 48 (1987), 235–242; Wei and Wylie, J. Differential Geom. 83 (2009), 377–405) or m = 1 $m=1$ (Wylie, Trans. Amer. Math. Soc. 369 (2017), 6661–6681; Wylie and D. Yeroshkin, Preprint). When m < 1 $m<1$ , our results are new in the literature.

Existence and Uniqueness of Green's Functions to Nonlinear Yamabe Problems

AbstractFor a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by‐product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Deformations of Q-curvature II

This is the second article of a sequence of research on deformations of Q-curvature. In the previous one, we studied local stability and rigidity phenomena of Q-curvature. In this article, we mainly investigate the volume comparison with respect to Q-curvature. In particular, we show that volume comparison theorem holds for metrics close to strictly stable positive Einstein metrics. This result shows that Q-curvature can still control the volume of manifolds under certain conditions, which provides a fundamental geometric characterization of Q-curvature. Applying the same technique, we derive the local rigidity of strictly stable Ricci-flat manifolds with respect to Q-curvature, which shows the non-existence of metrics with positive Q-curvature near the reference metric.

Comparison Theorems on Weighted Finsler Manifolds and Spacetimes with ϵ-Range

Abstract We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with ϵ-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie–Yeroshkin and Kuwae–Li.

Volume estimates for Alexandrov spaces with convex boundaries

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.

Comparison Geometry for an Extension of Ricci Tensor

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

A note on comparison theorems for graphs

In the present note we are concerned with the study of curvature-based comparison theorems on graphs related to main stochastic properties, such as the Feller property and stochastic completeness. We show that, under our main hypothesis, whilst previous results concerning stochastic properties are improved, it is not possible to obtain comparison theorems concerning volume growth. Finally, we prove an analogue of the Bishop-Gromov's relative volume comparison theorem and present a series of examples related to various possible notions of curvature.

Some Myers-Type Theorems and Comparison Theorems for Manifolds with Modified Ricci Curvature

In this paper we establish some new compactness criteria for complete Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below. These results improve or generalize previous ones obtained by H. Tadano [6], J. Wan [7], I.A. Kaboye and M. Bazanfar\'e [3]. We also prove a volume comparison theorem for such manifolds.

Comparison Theorems in Finsler Geometry with Integral Weighted Ricci Curvature Bounds and Their Applications

We establish the Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under integral weighted Ricci curvature bounds. As their applications, we obtain some results on integral weighted Ricci curvature and topology for Finsler manifolds. We also obtain some results on the upper bound of first eigenvalue for Finsler manifolds under integral weighted Ricci curvature bounds.

The Number of Cusps of Complete Riemannian Manifolds with Finite Volume

In this paper, we count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.

Bishop and Laplacian comparison theorems on Sasakian manifolds

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.

Density of Spherically Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum–minimum-distance codes. Computing a code’s density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.

Comparison Theorems and Their Applications on Kähler Finsler Manifolds

We obtain complex Hessian and Laplacian comparison theorems on Kahler Finsler manifolds with holomorphic bisectional curvature bounded from below. As applications, we prove Bishop type volume comparison theorem and Myers type theorem for Kahler Finsler manifolds. Moreover, we also obtain some comparison theorems for the first eigenvalue of Finsler–Laplacian and give a sharp upper estimate of the bottom of the spectrum for Kahler Finsler manifolds whose holomorphic bisectional curvature is bounded from below by $$-\,1$$.